Dispersionless Hierarchies, Hamilton-Jacobi Theory and Twistor Correspondences

The dispersionless KP and Toda hierarchies possess an underlying twistorial structure. A twistorial approach is partly implemented by the method of Riemann-Hilbert problem. This is however still short of clarifying geometric ingredients of twistor theory, such as twistor lines and twistor surfaces. A more geometric approach can be developed in a Hamilton-Jacobi formalism of Gibbons and Kodama. AMS Subject Classifiation (1991): 35Q20, 58F07,70H99